%% This document created by Scientific Word (R) Version 2.0


\documentclass[12pt,thmsa]{report}
\usepackage{amsfonts}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{sw20jrep}

%TCIDATA{TCIstyle=Report/report.lat,jrep,report}

%TCIDATA{Created=Sun Sep 01 18:03:51 1996}
%TCIDATA{LastRevised=Fri Sep 06 14:51:43 1996}
%TCIDATA{Language=American English}

\input tcilatex
\input tcilatex
\pagestyle{myheadings}
\markright{{\sf Causality and Dispersion Relations}}
\addtolength{\topmargin}{-0.4in}
\addtolength{\textheight}{1in}
\setcounter{secnumdepth}{0}
\setcounter{chapter}{10}
\thispagestyle{empty}
\begin{document}


\section{ 7.10 \quad Causality and Dispersion Relations.}

\subsection{7.10.1}

\quad \thinspace \textbf{a. }A frequency-dependent dielectric constant $%
\varepsilon (\omega )$ implies that $\mathbf{D}(t)$ does not depend just on $%
\mathbf{E}(t)$ at the same time $t,$ but that 
\begin{equation}
\mathbf{D}(t)=\int_{-\infty }^{+\infty }dt^{\prime }\varepsilon (t-t^{\prime
})\mathbf{E}(t^{\prime }),  \label{1}
\end{equation}
where 
\begin{equation}
\varepsilon (t-t^{\prime })=\int_{-\infty }^{+\infty }e^{-i\omega
(t-t^{\prime })}\varepsilon (\omega )\frac{d\omega }{2\pi }.\   \label{2}
\end{equation}
By the convolution theorem, (10.1) gives $\mathbf{D}(\omega )=\varepsilon
(\omega )\mathbf{E}(\omega ).$

$\smallskip\ $

\textbf{b.} Similar relations hold for other frequency-dependent quantities
such as $\chi (\omega )$, $\chi ^{mol}(\omega )$ (molecular polarizability), 
$\sigma (\omega ),\,n(\omega )$ (complex index of refraction). For instance,
the relation $\varepsilon (\omega )=1+4\pi \chi (\omega )$ translates to $%
\varepsilon (t-t^{\prime })=\delta (t-t^{\prime })+4\pi \chi (t-t^{\prime
}). $ The relation $\sigma (\omega )=-i\omega \chi (\omega )$ translates to 
\begin{equation}
\sigma (t-t^{\prime })=\frac{d}{dt}\chi (t-t^{\prime })  \label{3}
\end{equation}

$\smallskip\ $

\textbf{c. }As an example, suppose that 
\[
\varepsilon (\omega )=1+\omega _{p}^{2}/\left( \omega _{0}^{2}-\omega
^{2}-i\gamma \omega \right) . 
\]
Then 
\[
\varepsilon (\tau )=\delta (\tau )+\left( \omega _{p}^{2}/\nu _{0}\right)
\exp \left( -\gamma \tau /2\right) \sin (\nu _{0}\tau )\,\theta (\tau ), 
\]
where $\nu _{0}^{2}=\omega _{0}^{2}-\gamma ^{2}/4,\,\tau =t-t^{\prime }$ and 
$\theta (\tau )\,\,=1$ for $\tau >0,$ $\,\,0$ for $\tau <0.$ Thus $\chi
(t-t^{\prime })$ or $\sigma (t-t^{\prime })$ vanish for $t<t^{\prime },$
i.e. the polarization, or the current, are present only at times following
the application of the field. This describes mathematically the concept of
causality.

$\smallskip\ $

\textbf{d.} The property that $\chi (\tau )$, or $\sigma (\tau ),$ vanish
for $\tau <0$ is entirely general. If we can write $\chi (\omega )=N\chi
^{mol}(\omega ),$ it follows from the fact that the poles of $\chi
^{mol}(\omega )$ (see 7.5.2) are all in the lower half $\omega -$plane.
Adding a finite $\gamma >0,$ as in 7.5.4 c, does not change this.

\newpage

\subsection{7.10.2}

\quad \thinspace \textbf{a. }Vice-versa, if $\sigma (\tau )$ vanishes for $%
\tau <0,$ it follows that 
\begin{equation}
\sigma (\omega )=\int_{0}^{+\infty }\sigma (\tau )\exp \left( i\omega \tau
\right) d\tau  \label{4}
\end{equation}
is an analytic function of $\omega $ in the upper $\omega -$plane, provided
that $\sigma (\tau )$ is not exponentially divergent for $\tau \rightarrow
\infty .$ If $\sigma (\omega )$ approaches 0 faster than $\tau ^{-1},$ the
real $\omega $ axis is included in the region of analyticity of $\sigma
(\omega );$ otherwise it forms a boundary of this region.

$\smallskip\ $

\textbf{b. }The electric susceptibility $\chi (\omega )=i\sigma (\omega
)/\omega $ has analytic properties similar to $\sigma (\omega ),$ but can
have a pole on the real axis at $\omega =0$ if $\sigma (0)\neq 0,$ as in a
conductor. The dielectric constant $\varepsilon (\omega )=1+4\pi \chi
(\omega )$ can also have a pole at $\omega =0$ and in addition tends to $1$
(rather than $\omega $) for $\omega \rightarrow \infty .$ For these reasons
the analytic structure of $\sigma (\omega )$ is simpler and will be
discussed first.

$\smallskip\ $

\textbf{c.} It can be shown quite generally that $\sigma (\omega )$
approaches $iZNe^{2}/m\omega $ for $\omega \rightarrow \infty .$
Correspondingly, $\varepsilon (\omega )-1$ approaches $-\omega
_{p}^{2}/\omega ^{2},$ where $\omega _{p}$ is the plasma frequency. The
reason for this is that for high frequencies the electrons can be regarded
as free (quantum-mechanically, $\hbar \omega $ becomes larger than all the
relevant binding energies). Then each electron obeys $md\mathbf{v}/dt=e%
\mathbf{E}$ and we find $\mathbf{J=}$ $ZNe\mathbf{v=}$ $\left(
iZNe^{2}/m\omega \right) \mathbf{v}.$ This agrees, of course, with Drude's
formula (7.5.9) with $N_{c}=NZ,\,m^{*}=m$ and $\tau \rightarrow \infty .$

$\smallskip\ $

\textbf{d. }It follows that we can define, if we wish, the plasma frequency
as 
\begin{equation}
\omega _{p}^{2}=\lim_{\omega \rightarrow \infty }\{\omega ^{2}[1-\varepsilon
(\omega )\}.  \label{5}
\end{equation}

\newpage

\subsection{7.10.3}

\quad \thinspace \textbf{a. }It is clear from (\ref{4}) that $\func{Re}%
\sigma (\omega )$ and $\func{Im}\sigma (\omega )$ are not independent, since
they can both be derived from the single real function $\ \sigma (\tau ).$
The explicit relations between $\func{Re}\sigma $ and $\func{Im}\sigma $ can
be derived in a number of ways, from the analytic properties of $\sigma
(\omega )$ or directly from the definition (10.4). They are the
Kramers-Kronig dispersion relations: 
\begin{equation}
\text{$\func{Re}$}\sigma (\omega )=\frac{1}{\pi }\,\Bbb{P}\int_{-\infty
}^{+\infty }\frac{\text{$\func{Im}$}\sigma (\omega ^{\prime })}{\omega
-\omega ^{\prime }}d\omega ^{\prime }=\frac{2}{\pi }\,\Bbb{P}%
\int_{0}^{+\infty }\frac{\omega ^{\prime }\text{$\func{Im}$}\sigma (\omega
^{\prime })}{\omega ^{\prime \,2}-\omega ^{2}}d\omega ^{\prime }  \label{6}
\end{equation}
\begin{equation}
\text{$\func{Im}$}\sigma (\omega )=-\frac{1}{\pi }\,\Bbb{P}\int_{-\infty
}^{+\infty }\frac{\text{$\func{Re}$}\sigma (\omega ^{\prime })}{\omega
-\omega ^{\prime }}d\omega ^{\prime }=-\frac{2\omega }{\pi }\,\Bbb{P}%
\int_{0}^{+\infty }\frac{\text{$\func{Re}$}\sigma (\omega ^{\prime })}{%
\omega ^{\prime \,2}-\omega ^{2}}d\omega ^{\prime }  \label{6'}
\end{equation}

\smallskip\ 

\textbf{b. }In the last equality above we have used the fact that $\sigma
^{*}(\omega )=\sigma (-\omega ),$ or $\func{Re}\sigma (\omega )=$ $\func{Re}%
\sigma (-\omega )$ and $\func{Im}\sigma (\omega )=-\func{Im}\sigma (-\omega
).$ The symbol $P$ denotes the Cauchy principal part value.

\smallskip\ 

\textbf{c.} Similar relations hold for other response functions if we are
careful to ``subtract'' the parts that do not behave ``properly''. Thus the
relations hold for $\chi (\omega )=i\sigma (\omega )/\omega ,$ provided that 
$\sigma (0)=0;$ if $\sigma (0)\neq 0$ we must ``subtract'' $i\sigma
(0)/\omega .$ Keeping in mind that $\sigma (0)$ is real as discussed, we
have dispersion relations of the ``standard'' type between $\func{Re}\chi
(\omega )$ and $\func{Im}\chi (\omega )+\sigma (0)/\omega .$ The result is
given by Jackson (7.120) and problem 7.15 in terms of $\varepsilon (\omega
)-1,$ instead of $\chi (\omega ).$ The ``subtraction'' $-1$ is made
necessary by the fact that $\varepsilon \rightarrow 1$ for $\omega
\rightarrow \infty .$ We have then 
\[
\text{$\func{Re}\,$}\varepsilon (\omega )=1+\frac{2}{\pi }\,\Bbb{P}%
\int_{0}^{+\infty }\frac{\omega ^{\prime }\text{$\func{Im}\varepsilon $}%
(\omega ^{\prime })}{\omega ^{\prime \,2}-\omega ^{2}}d\omega ^{\prime }
\]
\[
\func{Im}\varepsilon (\omega )=\frac{4\pi \sigma (0)}{\omega }-\,\frac{%
2\omega }{\pi }\,\Bbb{P}\int_{0}^{+\infty }\frac{\func{Re}\varepsilon
(\omega ^{\prime })-1}{\omega ^{\prime \,2}-\omega ^{2}}d\omega ^{\prime }
\]

\smallskip\ 

\newpage

\subsection{7.10.4}

\quad \textbf{a. }The dispersion relations (\ref{6}) and (\ref{6'}) offer a
ready way to derive \textsl{sum rules}. By setting $\omega =0$ in (10.6), we
find 
\begin{equation}
\func{Re}\sigma \left( 0\right) =\frac{1}{\pi }\int_{-\infty }^{+\infty }%
\frac{\text{$\func{Im}$}\sigma (\omega ^{\prime })}{\omega ^{\prime }}%
\,d\omega ^{\prime }=\frac{1}{\pi }\int_{-\infty }^{+\infty }\text{Re}\chi
(\omega ^{\prime })\,d\omega ^{\prime },  \label{7}
\end{equation}
which relates the d.c. conductivity to the average of the electric
susceptibility, or of the atomic polarizability, over all frequencies. Note
that $\sigma \left( 0\right) $ is necessarily real because $\sigma (\omega
)=\sigma ^{*}(-\omega ).$ We can rewrite (\ref{7}) as 
\begin{equation}
\int_{-\infty }^{+\infty }\left[ \func{Re}\varepsilon (\omega )-1\right]
d\omega =4\pi ^{2}\sigma (0).  \label{7'}
\end{equation}
For an insulator, this implies that the average of $\func{Re}\varepsilon
(\omega )$ over all frequencies is unity. .

\smallskip\ 

\textbf{b.} The sum rule ``par excellence'' is obtained by setting $\omega
\rightarrow \infty $ in (\ref{6'}). Recalling from 7.10.2\thinspace c that $%
\sigma (\omega )$ approaches $iZNe^{2}/m\omega $ for $\omega \rightarrow
\infty ,$ we find
\begin{equation}
\frac{ZNe^{2}}{m}=\frac{2}{\pi }\int_{0}^{+\infty }\func{Re}\sigma (\omega
^{\prime })\,d\omega ^{\prime }  \label{8}
\end{equation}
or 
\begin{equation}
\frac{2}{\pi }\int_{0}^{+\infty }\omega \text{$\func{Im}$}\varepsilon
(\omega )d\omega =\frac{4\pi ZNe^{2}}{m}=\omega _{p}^{2}.  \label{8'}
\end{equation}
Using $\sigma ^{*}(\omega )=\sigma (-\omega ),$ we can also write (\ref{8})
as 
\begin{equation}
\frac{ZNe^{2}}{m}=\frac{1}{\pi }\int_{-\infty }^{+\infty }\sigma (\omega
)d\omega   \label{8''}
\end{equation}

\smallskip\ 

\textbf{c.} With $\varepsilon =1+4\pi N\chi ^{mol}$, it is seen that (\ref
{8'}) is equivalent to the dipole sum rule $\sum_{j}f_{ij}=Z$ for the
oscillator strengths, since $\sum_{i}\rho _{i}=1.$ We recall for convenience
the expression of $\chi ^{mol}$ that was given on page 7.5.2 of the notes,
eq. (2): 
\[
\chi ^{mol}=\frac{e^{2}}{2m}\sum_{ij}\rho _{i}\,\frac{f_{ij}}{\omega
_{ij}^{2}-\omega ^{2}-i\omega \eta } 
\]

\newpage

\subsection{7.10.5}

\quad \thinspace \textbf{a. }Dispersion relations follow from causality
alone for any response function that describes what happens to a system
after the application of the external stimulus, or the evolution of the
system from given initial conditions. It is not necessary for their validity
that Maxwell's or Newton's equations should hold. Hence they also apply when
the dynamical laws are not known (yet), as for instance in elementary
particle theory. They also apply to the evolution of a statistical ensemble
of systems, both in and out of thermal equilibrium.

\smallskip\ 

\textbf{b. }Sum rules always follow from dispersion relations, but the value
of the ``sum'' depends on knowledge of the behavior of the system in some
limit, usually $\omega =0$ or $\omega \rightarrow \infty .$ This may be
found from experiment (as $\sigma (0)\equiv \sigma _{d.c.})$ or be deducible
from gross properties of the system (such as $\omega _{p}^{2}).$

\smallskip\ 

\textbf{c. }Any good (and useful) approximate expression for the response
function should satisfy the dispersion relations and sum rules. It is
usually easier to check that there are no singularities in the upper
half-plane and that the behavior for $\omega \rightarrow \infty $ is
correct, rather than verify that the K-K formulae hold.

\smallskip\ 

\textbf{d.} The Drude expression for the conductivity, 
\[
\sigma (\omega )=\frac{Ne^{2}\tau }{m}\frac{1}{1-i\omega \tau } 
\]
satisfies the dispersion relations. If it is written as 
\[
\sigma (\omega )=\frac{\sigma _{d.c.}}{1-4\pi i\omega \sigma _{d.c.}/\omega
_{p}^{2}}, 
\]
it is clearly seen to be the simplest interpolation between $\sigma _{d.c.}$
at $\omega =0$ and the correct behavior at $\omega \rightarrow \infty .$

The one-oscillator with damping formula of 7.10.1\thinspace c and of next
section is also K-K correct.

\newpage

\subsection{7.10.6}

\quad \thinspace \textbf{a. }The ``dipole'' or ``oscillator strength'' or
``Thomas-Kuhn'' sum rule appears in many different disguises and is worth
commenting on. Refer back to 7.10.4 b and c.

\smallskip\ 

\textbf{b. }If an impulsive force $e\mathbf{E}\delta \mathbf{(}t^{\prime }%
\mathbf{)}$ is applied to a particle of charge $e,$ the velocity at time $%
t=0^{+}$ has increased by $\Delta \mathbf{v=}$ $e\mathbf{E/}m,$ regardless
of the interaction with other particles. WE can assume that at time $t=0^{-}$
the current is zero (at equilibrium). Then at time $0^{+}$ the current is $%
ZNe^{2}\mathbf{E}/m,$ which should be equated to $\sigma \left( \tau
=0^{+}\right) \mathbf{E.}$ It follows that
\begin{equation}
\frac{ZNe^{2}}{m}=\lim_{\tau \rightarrow 0^{+}}\int \frac{d\omega }{2\pi }%
\,\sigma (\omega )e^{-i\omega \tau }  \label{9}
\end{equation}
\begin{equation}
0=\lim_{\tau \rightarrow 0^{-}}\int \frac{d\omega }{2\pi }\,\sigma (\omega
)e^{-i\omega \tau }=\lim_{\tau \rightarrow 0^{+}}\int \frac{d\omega }{2\pi }%
\,\sigma (\omega )e^{i\omega \tau }.  \label{9'}
\end{equation}

\smallskip\ 

\textbf{c.} Adding (\ref{9}) and (\ref{9'}) side by side we find 
\begin{equation}
\frac{ZNe^{2}}{m}=\lim_{\tau \rightarrow 0^{+}}\frac{1}{\pi }\int_{-\infty
}^{+\infty }\sigma (\omega )\cos \left( \omega \tau \right) d\omega
\label{9''}
\end{equation}
which is equivalent to (\ref{8''}).

\smallskip\ 

\textbf{d.} The limits must be taken carefully. Using $\sigma (\omega
)=\sigma ^{*}(-\omega ),$ the r.h.s. of (\ref{9}) is 
\begin{equation}
\int_{-\infty }^{+\infty }\frac{d\omega }{2\pi }\left[ \text{Re}\sigma
(\omega )\cos \left( \omega \tau \right) +\text{$\func{Im}$}\sigma (\omega
)\sin \left( \omega \tau \right) \right]  \label{9'''}
\end{equation}
One cannot just set $\sin \left( \omega \tau \right) \rightarrow 0$ for $%
\tau \rightarrow 0^{+}$ and drop the second term in $[].$ For instance, for
free electrons $\func{Im}\sigma =ZNe^{2}/m\omega $ and 
\[
\int_{-\infty }^{+\infty }\frac{\sin \left( \omega \tau \right) }{\omega }%
\,d\omega =\pi \quad \quad \text{for \ }\tau >0 
\]
independent of $\tau .$ In fact the two terms of (\ref{9'''}) contribute
equally in (\ref{9}) and subtract in (\ref{9'})

\newpage

\subsection{7.10.7}

\quad \textbf{a. }From the discussion of previous page, it is seen that the
dipole sum rule results from the fact that the ``short-time'' response (or
equivalently the high-frequency response) is independent from interactions
between the particles.

\smallskip\ 

\textbf{b. }The generalizations of the sum rule all depend on this
possibility of neglecting the interactions. Referring back to
7.5.1\thinspace c, the relation between current and field in an isotropic
medium is 
\begin{equation}
\mathbf{J(r,}t)=\int_{all\,\,space}d^{3}r^{\prime }\int_{-\infty }^{+\infty
}dt^{\prime }\sigma \left( \mathbf{r,r}^{\prime };t-t^{\prime }\right) 
\mathbf{E}\left( \mathbf{r}^{\prime },t^{\prime }\right)   \label{10}
\end{equation}
If the medium is translationally invariant, any property such as $\sigma $
depends only on $\mathbf{r-r}^{\prime }.$ We introduce space and time
Fourier transforms according to the model 
\[
\mathbf{J(r,}t)=\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\int \frac{%
d\omega }{2\pi }\,\mathbf{J(k,}\omega )e^{i\left( \mathbf{k\cdot r}-\omega
t\right) }
\]

\smallskip\ 

\textbf{c. }By the convolution theorem, (\ref{10}) becomes simply 
\begin{equation}
\mathbf{J(k,}\omega )=\sigma \mathbf{(k,}\omega )\mathbf{E(k,}\omega )
\label{11}
\end{equation}
The dispersion relations and the sum rule hold for every $\mathbf{k,}$ and
it is true that $\sigma \mathbf{(k,}\omega )$ approaches $ZNe^{2}/m\omega $
for every $\mathbf{k,}$ as $\left| \omega \right| \rightarrow \infty .$

\smallskip\ 

\textbf{d.} The connection with the ``simple'' theory given in Jackson is
that the effective $\mathbf{k}$ in electromagnetic wave propagation is such
that $c\left| \mathbf{k}\right| =n(\mathbf{k,}\omega )\omega $. Since $c$ is
large, the effective $k$ is small (or the wavelength is usually large
compared to characteristic lengths in the medium). Then we can in fact set $%
\mathbf{k=}0.$

\newpage

\subsection{7.10.8}

\quad \thinspace \textbf{a.} A major application of the K-K relations occurs
in the analysis of experimental data. Suppose for instance that one wants to
measure the complex dielectric constant of a metal in the optical region
(from which all other optical properties can be computed). The steps are
outlined below.

\smallskip\ 

\textbf{b. }The easiest measurement to carry out is that of the reflectivity
at normal incidence as a function of $\omega .$ According to 7.3.5 c with $%
n=1$ for incidence from vacuum, the reflected electric field is given by 
\[
\frac{E_{0}^{\prime \prime }}{E_{0}}=\frac{n^{\prime }-1}{n^{\prime }+1}, 
\]
valid also for complex $n^{\prime }=\sqrt{\varepsilon ^{\prime }}.$

\smallskip\ 

\textbf{c. }The reflection coefficient, which is directly measurable, is $%
R=\left| E_{0}^{\prime \prime }/E_{0}\right| ^{2}.$ We see that to determine 
$\varepsilon ^{\prime }$ this is enough. However, we can put $E_{0}^{\prime
\prime }/E_{0}=\sqrt{R}\exp (i\theta )$, hence ln$\left( E_{0}^{\prime
\prime }/E_{0}\right) =\left( 1/2\right) \ln R+i\theta .$ Barring cases when 
$n^{\prime }=\pm 1,$ which can be handled by ``subtractions'' as necessary,
we see that ln$\left( \frac{n^{\prime }-1}{n^{\prime }+1}\right) $ is
analytic in the upper half plane, just as $n^{\prime }$ and $\varepsilon
^{\prime }$ are analytic there. Hence $\left( 1/2\right) \ln R$ and $\theta $
are K-K conjugates of each other.

\smallskip\ 

\textbf{d. }The procedure is then: measure $R$ as a function of $\omega $,
form $\left( 1/2\right) \ln R(\omega ),$ compute $\theta (\omega )$ by K-K
and find $n^{\prime }(\omega )$ from 
\[
n^{\prime }=\frac{1+\sqrt{R}\exp (i\theta )}{1-\sqrt{R}\exp (i\theta )}. 
\]
Note that $\ \theta $ is the phase lag upon reflection.

\newpage

\subsection{7.10.9}

\quad \thinspace \textbf{a. }While the K-K relations, which involve only
real $\omega $ and measurable quantities, are most useful in the direct
processing of experimental data, the analytic structure of or $\varepsilon
(\omega )$ is worth studying further to shed light into the structure of the
theory and obtain other relations.

\smallskip\ 

\textbf{b. }Using $\varepsilon (\omega )$ for conformity with Jackson and
supposing $\sigma (0)=0$ for simplicity, we start from the fact that for $z$
in the upper plane (Jackson (7.117)): 
\begin{equation}
\varepsilon (z)=1+\frac{1}{2\pi i}\int_{-\infty }^{+\infty }\frac{%
\varepsilon (\omega )-1}{\omega -z}d\omega .  \label{12}
\end{equation}

\smallskip\ 

\textbf{c.} By taking the limit of $z$ approaching the real axis, we find
the K-K relations (Jackson (7.119)). Now we can use these relations back in
the r.h.s. of (\ref{12}) to eliminate either $\func{Re}\varepsilon (\omega )$
or $\func{Im}$ $\varepsilon (\omega ).$ These results are: 
\begin{equation}
\varepsilon (z)=1+\int_{-\infty }^{+\infty }\frac{\text{$\func{Im}$}%
\varepsilon (\omega )}{\omega -z}\frac{d\omega }{\pi }  \label{13}
\end{equation}
\begin{equation}
\varepsilon (z)=1+\int_{-\infty }^{+\infty }\frac{\func{Re}\varepsilon
(\omega )}{\omega -z}\frac{d\omega }{\pi i}  \label{14}
\end{equation}
It is obvious that (\ref{12}) follows by adding (\ref{13}) and (\ref{14})
side by side; less obvious that any if these equations implies the
analyticity of $\varepsilon (z)$ and the other two equations.

\smallskip\ 

\textbf{d.} If we rewrite (\ref{13}) as 
\[
\varepsilon (z)=1+\frac{2}{\pi }\int_{0}^{+\infty }\frac{\omega ^{\prime }%
\text{$\func{Im}$}\varepsilon (\omega ^{\prime })}{\omega ^{\prime 2}-z^{2}}%
d\omega ^{\prime }, 
\]
set $z=\omega +i\eta $ and compare with (7.5.2), we can make the
identification 
\[
\frac{2}{\pi }\omega ^{\prime }\text{$\func{Im}$}\varepsilon (\omega
^{\prime })=Ne^{2}\sum_{ij}\rho _{i}f_{ij}\left[ \delta \left( \omega
^{\prime }-\omega _{ij}\right) -\delta \left( \omega ^{\prime }+\omega
_{ij}\right) \right] 
\]
in agreement with (7.5.3).

\newpage\ 

\subsection{7.10.10}

\quad \thinspace \textbf{a. }The gist of the argument on previous page is
that, when the system can be regarded as an assembly of independent
molecules, $\omega \func{Im}\varepsilon (\omega )$ (or Re$\sigma (\omega ))$
is directly related to the oscillator strength for optical transitions in
the frequency interval between $\omega $ and $\omega +d\omega $ of the
spectrum.

\smallskip\ 

\textbf{b.} Thus ($2/\pi )\omega \func{Im}\varepsilon (\omega )$ can be
properly called a spectral density and (\ref{13}) is called the spectral
representation of $\varepsilon .$ If $\sigma $ is used, ($2/\pi )$Re$\sigma
(\omega )$ is the spectral density.

\smallskip\ 

\textbf{c. }The result (\ref{13}) is entirely general and so is the
interpretation given above. The formula (7.5.5) is only approximate (unless $%
\gamma _{j}$ is allowed to depend on $\omega ),$ but the formula 
\begin{equation}
\varepsilon (\mathbf{k,}\omega )=1+4\pi \frac{e^{2}}{m}\sum_{n}\frac{f_{n}(%
\mathbf{k)}}{\omega _{n}^{2}-\omega ^{2}-i\omega \eta }  \label{15}
\end{equation}
is entirely general, if $\omega _{n}$ represent transitions frequencies for
the entire system and $f_{n}$ are corresponding oscillator strengths,
involving matrix elements of the current and density operators between the
exact states of the system in the absence of applied fields. In practice,
such states are not known (unless the system consists of ``small''
non-interacting parts, which we can call molecules, in which case (7.5.3) is
recovered). Therefore (\ref{15}) is replaced by some approximate expression
such as (7.5.5) if a connection with microscopic properties is desired.
However, we note again that the spectral density is directly measurable and
the representation (\ref{13}), which is equivalent to the K-K relations,
follows simply from causality.

\smallskip\ 

\textbf{d. }Other consequences of (\ref{13}) are in problem 7.16 of Jackson.
In particular, $1/\varepsilon $ has the same analytic properties as $%
\varepsilon .$ It follows from this that $\varepsilon (z)$ has no zeros for $%
\func{Im}z>0.$

\end{document}
